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arXiv · 2011.07481

Flips on homologous orientations of surface graphs with prescribed forbidden facial circuits

Weijuan Zhang·Jianguo Qian
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Abstract

Let $G$ be a graph embedded on an orientable surface. Given a class ${\cal C}$ of facial circuits of $G$ as a forbidden class, we give a sufficient-necessary condition for that an $\alpha$-orientation (orientation with prescribed out-degrees) of $G$ can be transformed into another by a sequence of flips on non-forbidden circuits and further give an explicit formula for the minimum number of such flips. We also consider the connection among all $\alpha$-orientations by defining a directed graph ${\bf D}({\cal C})$, namely the ${\cal C}$-forbidden flip graph. We show that if ${\cal C}\not=\emptyset$, then ${\bf D}({\cal C})$ has exactly $|O(G,{\cal C})|$ components, each of which is the cover graph of a distributive lattice, where $|O(G,{\cal C})|$ is the number of the $\alpha$-orientations that has no counterclockwise facial circuit other than that in ${\cal C}$. If ${\cal C}=\emptyset$, then every component of ${\bf D}({\cal C})$ is strongly connected. This generalizes the corresponding results of Felsner and Propp for the case that ${\cal C}$ consists of a single facial circuit.

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Weijuan Zhang, Jianguo Qian. 2020-11-15. Flips on homologous orientations of surface graphs with prescribed forbidden facial circuits. https://arxiv.org/abs/2011.07481

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